Quotients of Coxeter complexes and \(P\)-partitions.

*(English)*Zbl 0751.06002
Mem. Am. Math. Soc. 460, 134 p. (1992).

One aspect of this study involves finite Coxeter systems \((W,S)\), where \(W\) is a finite group generated by Euclidean reflections acting on a vector space of dimension \(| S|\), where \(S\) is the generating set of \(W\) and the associated Coxeter complex \(\Sigma(W,S)\) constructed in standard ways, for the study of which there exists a very considerable literature. In particular, if \(G<W\), then the orbit structure \(\Sigma(W,S)/G\), depending on the description of \(\Sigma(W,S)\) can be viewed as a topological or an algebraic object, but in any case one whose combinatorial nature is also an essential aspect to consider. This Memoir is based on the author’s doctoral thesis written under the supervision of R. Stanley and among its contributions it contains some interesting and deft extensions of ideas on \(P\)-partitions and multipartite \(P\)- partitions whose original investigation belongs to Stanley’s own Memoir volume based on his dissertation under G. C. Rota [R. P. Stanley, “Ordered structures and partitions”, Mem. Am. Math. Soc. 119 (1972; Zbl 0246.05007)]. Among the names associated with this entire circle connecting aspects of (combinatorial) topology with corresponding aspects in commutative-algebra-cum-algebraic-geometry and hence to the combinatorics-structure theory of posets, which, in turn, connects directly to order complexes and topology, dealing with shellability, Cohen-Macaulayness in the ordered portion of the theory, Stanley’s contributions touch on all facets in fundamental ways and his point of view is mirrored nicely in this particular work. All the standard actors appear and the circle remains unbroken in this discussion also, which succeeds in extending a considerable variety of important known results to new selections of pairs \(W\) and \(G\), e.g., reflection subgroups, in providing alternative and sometimes simpler proofs of results considered classical in this area along with new and interesting conclusions sometimes based on obviously useful notions such as “parsets”, for example. As a consequence, not only is it a useful addition to the literature on this subject, but an interested person can select this little volume as a good starting place whence to trace back to more detailed information in other monographs or to historical precedents on which results obtained here happen to be based even when altered and generalized. There is a sufficient though short bibliography while the introduction provides good orientation both to the material following and to the rationale for pursuing certain particular lines of reasoning.

Reviewer: J.Neggers (Tuscaloosa)